Let \(\mathrm{B}=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]\) and A be a \(2 \times 2\) matrix such that \(\mathrm{AB}^{-1}=\mathrm{A}^{-1}\). If \(\mathrm{BCB}^{-1}=\mathrm{A}\) and \(\mathrm{C}^{4}+\alpha \mathrm{C}^{2}+\beta \mathrm{I}=\mathrm{O}\), then \(2 \beta-\alpha\) is equal to :

JEE Main · 2024 · 09 Apr 2024 · Shift 2 · Matrices

Let

B = [1135]

and A be a

2 \times 2

matrix such that

AB^{- 1} = A^{- 1}

. If

BCB^{- 1} = A

and

C^{4} + α C^{2} + β I = O

, then

2 β - α

is equal to :

Answer: (D)

From

A B^{- 1} = A^{- 1}

, multiply both sides on the left by

A

A^{2} B^{- 1} = I ⟹ A^{2} = B

.

From

B C B^{- 1} = A

, squaring both sides:

(B C B^{- 1})^{2} = A^{2} ⟹ B C^{2} B^{- 1} = A^{2} = B

; pre-multiplying by

B^{- 1}

and post-multiplying by

B

gives

C^{2} = B

.

So

C^{2} = B = (1135)

. The characteristic equation of

B

λ^{2} - 6 λ + 2 = 0

(trace

= 6

, det

= 5 - 3 = 2

), so by Cayley–Hamilton,

B^{2} - 6 B + 2 I = O

.

Substituting

B = C^{2}

C^{4} - 6 C^{2} + 2 I = O

.

Comparing with

C^{4} + α C^{2} + β I = O

α = - 6, β = 2

2 β - α = 4 - (- 6) = 10

JEE Main · 2024 · 09 Apr 2024 · Shift 2 · Single correct option